Rational elliptic curves and holomorphic modular forms

In 1985-1986, Frey, Serre and Ribet reduced the proof of the famous Fermat's Last Theorem to a proof of the Taniyama-Weil-Shimura Conjecture for semistable elliptic curves. In 1995 Wiles proved the Taniyama-Weil-Shimura Conjecture for semistable elliptic curves, thus completing the proof of the Fermat's Last Theorem.;It is believed that the popularity of the Fermat's Last Theorem is due basically to its very short and easy statement. However, a complete statement of the Taniyama-Weil-Shimura Conjecture is not short at all. The goal of this thesis is to supply almost all of the definitions necessary in order to make a clear and complete statement of this conjecture, one of the major conjectures in number theory.